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If X = {−∞}, then sup X = −∞. Write down the similar statements for inf X. 37. For a not necessarily bounded sequence {xn }, extend the definition of LIM{xn } by adding +∞ if there is a subsequence diverging to +∞, and adding −∞ if there is a subsequence diverging to −∞. 36. 11. 1. A sequence with no upper bound must have a subsequence diverging to +∞. This means limn→∞ xn = +∞. 2. If there is no subsequence with finite limit and no subsequence diverging to −∞, then the whole sequence diverges to +∞.